Single station system and method of locating lightning strikes

ABSTRACT

An embodiment of the present invention uses a single detection system to approximate a location of lightning strikes. This system is triggered by a broadband RF detector and measures a time until the arrival of a leading edge of the thunder acoustic pulse. This time difference is used to determine a slant range R from the detector to the closest approach of the lightning. The azimuth and elevation are determined by an array of acoustic sensors. The leading edge of the thunder waveform is cross-correlated between the various acoustic sensors in the array to determine the difference in time of arrival, ΔT. A set of ΔT&#39;s is used to determine the direction of arrival, AZ and EL. The three estimated variables (R, AZ, EL) are used to locate a probable point of the lightning strike.

This application claims the benefit of provisional application No.60/217,601, filed Jul. 11, 2000.

ORIGIN OF THE INVENTION

The invention described herein was made in the performance of work undera NASA contract and is subject to the provisions of Section 305 of theNational Aeronautics and Space Act of 1958, as amended, Public Law85-568 (72 Stat. 435; 42 U.S.C. §2457).

TECHNICAL FIELD OF THE INVENTION

The present invention relates generally to identifying locations oflightning strikes.

BACKGROUND OF THE INVENTION

Electronic equipment is susceptible to damage caused by nearby lightningstrikes. The accurate knowledge of a lightning striking point isimportant to determine which equipment or system needs to be testedfollowing a lightning strike. Existing lightning location systems canprovide coverage of a wide area. For example, a lightning locationsystem can provide coverage of an area having a 30 km radius. Thissystem, however, has a 50% confidence region of about 500 meters. Thatis, the system has a 50% confidence that a lighting strike is within 500meters of an identified location. As such, present lightning locationsystems cannot be used to determine whether a lightning strike occurredinside or outside of a parameter of an area of concern. One suchapplication of a lightning location system is a Space Shuttle launch padfor the National Aeronautics and Space Administration (NASA). Byaccurately determining lightning strike locations, electronic equipmentlocated within the launch pad area can be tested and/or reset to avoiderroneous operation.

One method of determining the location of lightning strikes uses a setof video cameras that are pointed in different directions within thearea of concern. If a lightning strike occurs within the field of viewof three or more cameras, the location of the strike can be determined.However, if the cameras are not pointed in the correct direction, oreither an object or a heavy rain downpour obscures their field of view,it is difficult or impossible to accurately determine a striking pointof the lightning. Further, this method has a relatively largeuncertainty and does not facilitate an accurate location of the exactpoint of contact to the ground.

For the reasons stated above, and for other reasons stated below whichwill become apparent to those skilled in the art upon reading andunderstanding the present specification, there is a need in the art forthe system and method to accurately approximate locations of lightningstrikes.

SUMMARY OF THE INVENTION

The above-mentioned problems with lightning strike location and otherproblems are addressed by the present invention and will be understoodby reading and studying the following specification.

In one embodiment, a system to determine an approximate location oflighting strikes comprises an electric field sensor, a plurality ofacoustic sensors collocated with the electric field sensor, and aprocessor to determine the location of lighting strikes based on a timedifferential between an electric field pulse and a acoustic wave.

In another embodiment, a system to determine an approximate location oflighting strikes comprises an electric field sensor, a plurality ofacoustic sound sensors collocated with the electric field sensor, atemperature sensor, and a processor. The processor determines thelocation of lighting strikes based on a time differential between anelectric field pulse and a acoustic wave. The processor uses a look-uptable to compensate for ambient temperature measured by the temperaturesensor.

A method for determining an approximate location of a lightning strikecomprises collecting lightning strike information, including a time ofarrival of an electric field pulse using an electric field sensor andtime of arrival data of an associated sonic wave from a lightningstrike. The sonic wave data is detected using a plurality of acousticsensors collocated with the electric field sensor, and processing thelightning strike information to approximate the location of thelightning strike.

Another method for determining an approximate location of a lightningstrike comprises detecting an electric field signal, detecting a sonicpulse with a plurality of collocated sensors, measuring a time ofarrival difference between the electric field signal and the sonicpulse, and determining a range, azimuth and elevation of the lightingstrike.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates one embodiment of a single lightning location system;

FIG. 2 illustrates a shape of a wave front distorted by wind;

FIG. 3 illustrates a baseline and hyperbola;

FIG. 4 illustrates a baseline and hyperbola with an infinite source;

FIG. 5 illustrates a baseline and a location cone;

FIG. 6 illustrates a definition of AZ and EL;

FIG. 7 shows a close-up of three intersecting lines of position;

FIG. 8 is a conceptual diagram of an error ellipse; and

FIG. 9 is a flow chart of one method of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

In the following detailed description of the preferred embodiments,reference is made to the accompanying drawings that form a part thereof,and in which is shown by way of illustration specific preferredembodiments in which the present invention may be practiced. Theseembodiments are described in sufficient detail to enable those skilledin the art to practice the present invention, and it is to be understoodthat other embodiments may be utilized and that logical, mechanical andelectrical changes may be made without departing from the spirit andscope of the present invention. The following detailed description is,therefore, not to be taken in a limiting sense, and the scope of theinvention is defined only by the claims.

The present invention provides a system and method that can accuratelylocate a lightning strike within an area of interest. In one embodiment,the present invention can accurately locate a lightning strike within afew meters. The fast varying electric current associate with lightningdischarges generates large electric field variations. The electric fieldwaveform propagates at the speed of light in a radial direction from thestriking point of lightning. The sudden heating of the air caused by thelarge currents associated with the lightning discharge produces a suddenexpansion of the air near the lightning channel. This results in a sonicwave (thunder) that initially, for the first few meters, propagates at asupersonic speed and later propagates at a sonic speed.

For an observer located remotely from a lightning strike location, theelectric field waveform arrives earlier than the sonic sounds. This isbecause the electric field waveform travels and a speed of approximately300,000,000 m/s, while the sonic wave travels and approximately 350 m/s.An observer can determine the distance to the striking point bymeasuring the time difference between the arrival of the electric fieldwaveform and the arrival of the sonic wave. This measurement defines acircle, with the observer at the center, in which the lightning strikemight have occurred. A second observer at a different location using thesame type of measurement also has a circle defined around them in whichthe lightning might have occurred. These two circles intersect at twopoints. With the addition of a third observer, a single striking pointcan be determined.

In an embodiment described in U.S. patent application Ser. No.09/784,405 filed Feb. 13, 2001 now U.S. Pat. No. 6,420,862, a lightninglocator uses several (at least three) remote receiver stations eachequipped with a broad band antenna and a microphone. The antenna detectslightning through its strong electromagnetic pulse and starts a timer.The timer is turned off by a detection of the arrival of the sound wavedue to thunder. Range is estimated by multiplying the time delay by thespeed of sound. The ranges detected by each receiver are transmitted toa central processor to estimate the location of the lightning strike.

An embodiment of the present invention uses a single detection system.This system is triggered by a broadband RF detector and measures a timeuntil the arrival of a leading edge of the thunder acoustic pulse. Thistime difference is used to determine a slant range R from the detectorto the closest approach of the lightning channel. The azimuth andelevation (AZ and EL) are determined by an array of acoustic sensorswith one to several meter baselines. The leading edge of the thunderwaveform is cross-correlated between the various microphones in thearray to determine the difference in time of arrival (DTOA or ΔT). A setof ΔT's is used to determine the direction of arrival (AZ and EL). Thethree estimated variables (R, AZ, EL) are used to locate a probablepoint of the lightning strike.

The present invention provides a lightning location system that uses asingle station to determine an approximate location of lightningstrikes. One embodiment of the present location station 100 includes anelectric field sensor 102 and five sonic all receivers 104 a-e, see FIG.1. Four sonic sensors 104 a-d are located in a periphery of a circle at90 degrees from each other. For example, the sensors can be placed on a5-meter radius circle. The electric field sensor 102 is located at thecenter of the circle. The fifth sonic sensor 104 e is located above theplane formed by the four sensors, such as 2 meters above the plane. Thestation, therefore, resembles a pyramid or a conical shape. A processor106 is coupled to the sonic and electric field sensors to control thestation and process data. As explained herein, wind speed sensor 108,temperature sensor 110 and a humidity sensor 112 can be provided tocorrect the data processing and reduce errors. A camera 114 can also beadded to provide a visual reference during operation. The output fromthe system can provide both data output and visual output.

In operation, the electric field sensor receives an electric fieldwithin a few microseconds after a lighting strike. A sonic wave arrivesat the sonic sensors following the electric field. The sonic fieldreaches each sensor within milliseconds of each other, since thedistance to the striking point is different for each sonic sensor. Basedon the differences in the time of arrival of the sonic wave at eachsensor, a set of equations is solved to determine the angle of arrivalof the sonic waveform. The differences in the time of arrival can beprecisely measured by performing digit cross-correlations among thesignals received by the five sensors. The distance to the lightingstrike is determined by the time elapsed from the detection of theelectric field to the detection of the thunder signal (sonic). The speedof sound can be estimated by monitoring the ambient temperature andautomatically compensating for any deviations during the course of theday. A look-up table is used to determine the correct sound speed to usefor the calculation of the distance to the lightning strike.

Digital signal processing techniques are used to discriminate betweenthunder waveforms from a close lightning strike (within about a 1-mileradius) and thunder waveforms from a preceding distant strike. Thisprocess is important since, during the active phase of a thunderstorm,lightning activity could occur at a rate exceeding 10 strikes perminute. The discrimination between close and distant thunder is based onthe frequency content of the thunder waveform. Propagation through theair causes the high-frequency components to attenuate much faster thanthe lower frequencies. Thunder generated within short distances hastypical high frequency components (a “clap” of initial sound) followedby the longer duration rumble. Thunder from more distant waveforms losetheir high-frequency components and no “clap” sound is heard.

A designer of a single station sonic lightning location system canselect the geometry of the sonic sensor microphones in a variety ofmanners depending on the portion of the sky that is of greatestinterest. The analysis provided herein allows a designer to calculatethe accuracy of direction of arrival estimates based on user providedlocations of the sensor microphones and the timing accuracy possible.The timing accuracy is based on the electronic and digital design of thedetection system and also on the nature of the thunder waveformsactually experienced under field conditions. The math presented providesthe estimated azimuth and elevation of the source and estimates of theconfidence levels of these estimates (error ellipse).

The practically instantaneous arrival of the electromagnetic pulse isdetected with the broad band antenna 102 which starts a timer in theprocessor 106. When the thunder signal arrives at a selected microphone104, sonic sensor, the timer is stopped. A variety of techniques fordigital signal processing can be used to start and stop the timer andwill not be addressed here. A signal analysis technique is used toselect only nearby events by filtering, see U.S. application Ser. No.09/784,405 filed Feb. 13, 2001. Any known technique can be used toestimate a time delay of arrival of a sonic or thunder pulse accurate towithin a few microseconds. This time delay is multiplied by c, the speedof sound, to estimate the slant range to the lightning attach point(“ground” contact). With the speed of sound of roughly 300 meters persecond, this provides estimation accuracy on the order of a centimeteror less. However, the variation of the speed of sound with temperatureand humidity and the effects of wind degrade this ideal performance.

Wind velocities are significantly lower than sonic velocity. For examplea wind speed of 50 miles per hour equates to 22.6 meters per second,only 7.5% of the speed of sound (approximately 300 m/s). The thunderleaves the lightning channel as a cylindrical wave front with a velocitythat is the vector sum of the wind speed and the sonic speed. Thisaffects the overall travel time to the sensor, used to estimate theslant range, and potentially, the curvature of the wave front at thesensor affects the estimated azimuth and elevation (for the singlestation system). Assuming that the lightning channel is vertical and thewind velocity vector is strictly horizontal, any effects to theestimated elevation angle can be ignored. This reduces the problem totwo dimensions.

As shown in FIG. 2, a lightning location point is given by P and thesensor location is at the origin O. The wind velocity (to the south inthe figure) distorts the wave front by adding the components of the windvelocity to each wave front velocity vector. In the presence of wind,the velocity vector of the wave front measured along an azimuth θ isgiven by:

{right arrow over (V)}(θ)={right arrow over (c)}−|{right arrow over (V)}_(W)|cos(θ−∂)  (1)

The second term of equation (1) measures the contribution of the windalong the line of sight to the event P. The minus sign comes fromconsideration of the angles involved. Note that when ∂ is greater than θ(as in FIG. 2) the sign of the cosine term changes to plus increasingthe effective wavefront velocity. Thus, in the presence of wind, thewave front arrives sooner or later than it would in still air.Designating “ΔT” as the time delay between the lightning flash and thethunder arrival and recognizing that wind has no effect on theelectromagnetic propagation time, the range is:

R=ΔT·V=ΔT·(c−V _(W) cos(θ−∂))  (2)

This equation must be inverted so that the measurement ΔT is expressedin terms of unknown variables: $\begin{matrix}{{\Delta \quad T} = \frac{R}{\left( {c - {V_{W}\quad \cos \quad \left( {\theta - \vartheta} \right)}} \right)}} & (3)\end{matrix}$

The value of c changes with temperature and humidity (molecular weightof the atmosphere). The value of sound speed in a gas is given by:

c=(γR ₀ T/M)^(1/2)  (4)

where γ is the ratio of specific heats, R₀ is the universal gas constant(8314 J kg⁻¹K⁻¹), T is the absolute temperature and M is the molecularweight. The speed of sound varies as the square root of the temperature.The speed of sound increases by 0.6 m/s for each degree increase inCelsius temperature. This significant effect should be corrected usingtemperature sensor 110. The presence of water vapor causes both γ and Mto decrease. The water vapor effect can often be ignored since itscorrection is typically less than 1.5%. However, if humidity data isavailable to the system it would be advisable to correct for itseffects. To preserve accuracy, an external measurement of local windspeed and temperature may be desired. Measuring humidity provides asmaller benefit. If such measurements are provided the range accuracy ofthe present single station system is quite high.

Any two sonic receivers (R) form a “baseline,” that is a line connectingthe two receiver coordinates. The location of a source S that produces asignal lies on a hyperbola in the plane formed by two microphones andthe lightning source, see FIG. 3. For each point on a hyperbola, thedifference between the distances to the two foci is a constant. Eachmicrophone forms a focus of the hyperbola. This distance difference isthe difference time of arrival (DTOA), in time units, times the speed ofsound c. The location of S is unknown and, hence, the orientation of theplane, S lies on a “hyperbola of revolution” or a hyperbolic cone givenby rotating the hyperbola around the axis defined by the baseline.

The curve of a hyperbola, away from the origin, asymptoticallyapproaches a straight line that deviates from the baseline with interiorangle ψ, see FIG. 4. In the present invention, only distant pointsrelative to the length of the baselines are used. Therefore, theequations are simplified by using only straight lines instead ofhyperbolic curves. When the straight line is rotated about the axis, itgenerates a circular cone (FIG. 5).

The present system measures a time difference in microseconds betweenthe time of signal arrival at one microphone minus the time of arrivalto a second microphone. This DTOA times c, the speed of sound, is adistance difference. The measured TOA from microphone 1 is T1, and themeasured TOA from microphone 0 is T0 then:

ΔT ₀₁ =T 0−T 1  (5)

If the ΔT₀₁ term is positive, then T0 is a larger number and is later intime; the event reached “1” first. Therefore, the cone angles towardmicrophone 1. Likewise, if ΔT₀₁ is negative, the event reached “0” firstand the cone opening lies toward microphone 0 (ψ>π/2). The cone ofconstant ΔT₀₁ forms a constant angle between the baseline and the conesurface ψ (FIG. 2). In terms of the ΔT₀₁, for sources far away from thebaseline:

cΔT ₀₁ =l ₀₁ cos ψ  (6)

So that the cone angle ψ is given by: $\begin{matrix}{\psi = {{arc}\quad \cos \quad \left( {\frac{c}{l_{01}}\quad \left( {{T0} - {T1}} \right)} \right)}} & (7)\end{matrix}$

This cone traces a circular arc on the sky. Using multiple baselines canbe used to produce multiple arcs, such that their intersection definesthe azimuth and elevation of the source. In order to accomplish this,the baseline orientation is defined in terms of azimuth and elevation,which, coupled with the above DTOA equation, allow the definition of thearc. The AZ and EL angles can be defined in terms of local Cartesiancoordinate system. The orientation of each receiver pair baseline andthe direction to each source in terms of AZ and EL is expressed asfollows. The locally flat Cartesian coordinate system is called (east,north, up) for (x, y, z) respectively. The azimuth AZ and elevation ELangles are defined with reference to FIG. 5. Assuming that the referencereceiver (point 0) lies at the point (x0 ,y0, z0)=(0, 0, 0). Thebaseline vector is defined as the vector from (0, 0, 0) to the receiverlisted first in the DTOA equation. So that the baseline “10” vector runsfrom the point (0, 0, 0) to (x1, y1, z1). This vector defines a baselineorientation in terms of azimuth and elevation. The azimuth is measuredfrom the north (y axis) in a clockwise direction. The elevation angle ismeasured up from the (x, y) plane. Points lying in the (x, y) planedefine position vectors with a zero elevation angle.

The baseline vector is therefore given by:

{right arrow over (l)} ₁₀=(x ₁−0)î+(y ₁−0)

ĵ+(z ₁−0){circumflex over (k)}  (8)

The length of this vector is given by:

$\begin{matrix}{{{{\overset{\rightharpoonup}{l}}_{10}} = {\sqrt{x_{1}^{2} + y_{1}^{2} + z_{1}^{2}} = l_{10}}},} & (9)\end{matrix}$

where l₁₀, l₂₀ and l₃₀ indicate the length of three baselines vectors.It must be kept in mind that the results below will depend on the signof the DTOA, which is implicitly affected by the direction assumed forthe baseline vector. Given a point (x_(o), y_(o), z_(o)), which could bean antenna or a source point, it is transformed to spherical coordinatesas follows, where again l denotes the length of the vector from theorigin to the point, see FIG. 5:

z _(o) =l sin EL

x _(o) =l sin AZ=l cos EL sin AZ  

y _(o) =l cos AZ=l cos EL cosAZ  (10)

Since we are only interested in AZ, EL, we divide through by the lengthand solve for the angles: $\begin{matrix}{{EL} = {{arc}\quad \sin \quad \left( \frac{z_{O}}{l} \right)}} & \text{(11a)} \\{{AZ} = {{arc}\quad \tan \quad \left( \frac{y_{O}}{x_{O}} \right)}} & \text{(11b)}\end{matrix}$

The angles of the baseline vector can be defined using receiver 0 as thereference origin (0, 0, 0):${EL}_{10} = {{arc}\quad \sin \quad {\left( \frac{z_{1}}{\sqrt{x_{1}^{2} + y_{1}^{2} + z_{1}^{2}}} \right)}}$

$\begin{matrix}{{AZ}_{10} = {{arc}\quad \tan \quad \left( \frac{y_{1}}{x_{1}} \right)}} & (12)\end{matrix}$

Equation (12) defines the AZ and EL of any point in terms of ourCartesian system.

The following section derives the equation for the arc on the sky basedon a measured delta T, baseline length, and baseline orientation, and isa general treatment which can be solved for any baseline orientation.Given a source point S(x_(s), y_(s), z_(s)) or, in spherical polarcoordinates, (l_(s), AZ_(s), El_(s)) and a baseline unit vector:$\begin{matrix}\left( {\frac{c}{l_{01}}\quad \left( {{T0} - {T1}} \right)} \right) & (13)\end{matrix}$

Expressing the vector pointing from the origin to the source point:

{overscore (s)}=x _(s) î+y _(s) ĵ+z _(s) {circumflex over (k)}  (14)

and its corresponding unit vector: $\begin{matrix}{\hat{s} = {{\frac{x_{s}}{r_{s}}\hat{i}} + {\frac{y_{s}}{r_{s}}\hat{j}} + {\frac{z_{s}}{r_{s}}\hat{k}}}} & (15)\end{matrix}$

Note that the dot product between these two vectors is equal to thecosine of the angle between them.

ŝ·{circumflex over (l)}=cos ψ  (16)

The dot product is given by the sum of the products of the threecomponents: $\begin{matrix}{{\hat{s} \cdot \hat{l}} = {{\frac{x_{S}}{r_{S}} \cdot \frac{x_{1}}{r_{1}}} + {\frac{y_{S}}{r_{S}} \cdot \frac{y_{1}}{r_{1}}} + {\frac{z_{S}}{r_{S}} \cdot \frac{z_{1}}{r_{1}}}}} & (17)\end{matrix}$

Using the following spherical coordinate conversion formulas (equation10) to express the dot product in terms of angles:

ŝ·{circumflex over (l)}=cos EL _(s) sin AZ _(s) cos EL ₁ sin AZ ₁+cos EL_(s) cos AZ _(s) cos EL ₁ cos AZ ₁+sin EL _(s) sin EL ₁  (18)

Rearranging:

ŝ·{circumflex over (l)}=cos EL _(s) cos EL ₁ sin AZ _(s) sin AZ ₁+cos EL_(s) cos EL ₁ cos AZ _(s) cos AZ ₁+sin EL _(s) sin EL ₁  (19)

Using $\begin{matrix}{{{\cos \quad \psi} = \left( {\frac{c}{l_{01}}\quad \left( {{T0} - {T1}} \right)} \right)},} & \left( {{equation}\quad 7} \right)\end{matrix}$

the equations for an arbitrary baseline becomes: $\begin{matrix}\begin{matrix}{{\frac{c}{l_{01}}\quad \left( {{T0} - {T1}} \right)} = \quad {{\cos \quad {EL}_{S}\quad \cos \quad {EL}_{1}\quad \sin \quad {AZ}_{S}\quad \sin \quad {AZ}_{1}} +}} \\{\quad {{\cos \quad {EL}_{S}\quad \cos \quad {EL}_{1}\quad \cos \quad {AZ}_{S}\quad \cos \quad {AZ}_{1}} +}} \\{\quad {\sin \quad {EL}_{S}\quad \sin \quad {EL}_{1}}}\end{matrix} & (20)\end{matrix}$

Each baseline in the system produces an equation 20. Solving the set ofequations for an array of baselines is addressed below. In order tosolve for the azimuth and elevation of a distant source, at least threebaselines are needed. The lines of position provided by each baselinedescribe latitude type lines (i.e. not great circles) across the sphereof the sky. Two baseline-derived lines of position (LOP's) intersect attwo points. A third baseline is required to resolve the ambiguity.

Four sensors are needed to obtain three baselines. Care must be taken inselecting the baselines produced by the four sensors. In particular,three independent sets of baselines must be chosen such that onebaseline cannot be expressed as the difference between two others. Thiscauses the matrix equations to have a lower rank than three and be underdetermined. This effect prevents production of three baselines fromthree sensors. Beyond this constraint, however, the system can actuallyselect the arrangement of baselines to provide the best solution foreach particular event, if the system geometry allows it. Therefore, allof the solutions are over-determined in the sense that there are moreequations than unknowns. There may also be constraints such as theground level that can restrict solutions to portions of the sky (so thatelevation angles can only be between zero and positive pi over two).

Since each baseline measurement contains errors, it is desired to findthe solution AZ and EL point that minimizes the sum of the squareerrors. The solution point is called (AZ_(EST), EL_(EST)) and using thispoint the expected ΔT's for each baseline can be computed:$\begin{matrix}\begin{matrix}{{\Delta \quad T_{i}} = \quad {\frac{l_{i}}{c}\quad \left( {{\cos \quad {EL}_{s}\quad \cos \quad {EL}_{i}\quad \sin \quad {AZ}_{s}\quad \sin \quad {AZ}_{i}} +} \right.}} \\{\quad {{\cos \quad {EL}_{s}\quad \cos \quad {EL}_{i}\quad \cos \quad {AZ}_{s}\quad \cos \quad {AZ}_{i}} +}} \\{\quad \left. {\sin \quad {EL}_{s}\quad \sin \quad {EL}_{i}} \right)}\end{matrix} & (17)\end{matrix}$

The angles that involve the i^(th) baselines can be defined as constantsas follows:

α_(i)=cos EL _(i) sin AZ _(i), β₁=cos EL _(i) cos AZ _(i), γ_(i)=sin EL_(i)  (18)

With cos EL equal to CEL and sin AZ is SAZ, these changes simplifyequation 17 to: $\begin{matrix}{{{\Delta \quad T_{i}} = {\frac{l_{i}}{c}\left( {{\alpha_{i}{CEL}\quad {SAZ}} + {\beta_{i}{CEL}\quad {CAZ}} + {\gamma_{i}{SEL}}} \right)}}\quad} & (19)\end{matrix}$

A nonlinear regression on this set of equations produces a least squaresestimate of the (AZ, EL) of the source. An initial very rough estimateof the source location must first be produced. This can be done inseveral ways but a method is recommended whereby the closest grouping ofthree intersections is identified. This can be accomplished by creatinga table of computed delta T's for each 10 degree by 10−degree patch ofsky. Since AZ runs from 0 to 360 degree and EL runs from 0 to 90degrees, this results in a listing of 324 delta T sets (3 values each).The actual delta T's can then be subtracted from each computed delta Tthen squared. By finding the smallest value in this list, an estimate isprovided of the correct value. Using this as a starting value, the deltaT functions can be extended in a Taylor series around the estimatingstarting point. By using only the first order terms of the series, a setof linear equations is obtained as follows: $\begin{matrix}{{\Delta \quad T_{{Measured}\quad i}} = {{\Delta \quad T_{{Estimated}\quad i}} + {\frac{{\partial\Delta}\quad T_{i}}{\partial{AZ}_{Est}}\quad \left( {\Delta \quad {AZ}} \right)} + {\frac{{\partial\Delta}\quad T_{i}}{\partial{EL}_{Est}}\left( {\Delta \quad {EL}} \right)}}} & (20)\end{matrix}$

The partial derivative terms are computed as follows: $\begin{matrix}{\frac{\partial T_{i}}{\partial{AZ}_{Est}} = {\frac{l_{i}}{c}\quad \left( {{\alpha_{i}{CEL}_{Est}\quad {CAZ}_{Est}} - {\beta_{i}{CEL}_{Est}\quad {SAZ}_{Est}}} \right)}} & \text{(21a)}\end{matrix}$

$\begin{matrix}\begin{matrix}{\frac{\partial T_{i}}{\partial{EL}_{Est}} = \quad {\frac{l_{i}}{c}\quad \left( {{{- \alpha_{i}}{SEL}_{Est}\quad {SAZ}_{Est}} - {\beta_{i}{SEL}_{Est}\quad {CAZ}_{Est}} +} \right.}} \\{\quad \left. {\gamma_{i}{CEL}_{Est}} \right)}\end{matrix} & \text{(21b)}\end{matrix}$

Since the values of all of the parameters are known, these terms reduceto numbers for any particular choice of estimated AZ and EL position.With 3 (or more) baselines, a matrix equation is constructed which islinear in the differences in the delta T's and the corrections to AZ andEL. With:

δΔT _(i) ≡ΔT _(Measured i) −ΔT _(Estimated i)  (22)

the matrix equation can be written as: $\begin{matrix}{\begin{pmatrix}{\delta \quad \Delta \quad T_{1}} \\{\delta \quad \Delta \quad T_{2}} \\{\delta \quad \Delta \quad T_{3}}\end{pmatrix} = {\begin{pmatrix}\frac{\partial T_{1}}{\partial{AZ}_{Est}} & \frac{\partial T_{1}}{\partial{EL}_{Est}} \\\frac{\partial T_{2}}{\partial{AZ}_{Est}} & \frac{\partial T_{2}}{\partial{EL}_{Est}} \\\frac{\partial T_{3}}{\partial{AZ}_{Est}} & \frac{\partial T_{3}}{\partial{EL}_{Est}}\end{pmatrix}\quad \begin{pmatrix}{\Delta \quad {AZ}} \\{\Delta \quad {EL}}\end{pmatrix}}} & (23)\end{matrix}$

This can be written in matrix vector notation as:

δΔT=HΔΘ  (24)

This is solved using the generalized inverse (multiply both sides by thetranspose of matrix H then take the inverse of the now square H^(T)H).The solution is written as follows:

ΔΘ=(H ^(T) H)⁻¹ H ^(T) δΔT  (25)

The values of delta AZ and delta EL are added to the original estimatedvalues of AZ and EL to produce a new estimate. The values of the squaredδΔT terms are summed. The process is repeated with the new estimateuntil the sum of the squared residuals no longer decreases. Thisrepresents an un-weighted least squares estimate.

An optimal solution would weight each baseline based on the inherentaccuracy provided. Returning to the original equation for the delta T asa function of cosine of the angle between the baseline and the directionto the source: $\begin{matrix}{{\Delta \quad T} = {\frac{l}{c}\quad \cos \quad \psi}} & (26)\end{matrix}$

and taking the derivative of both sides (leaving l constant of course):$\begin{matrix}{{{\Delta}\quad T} = {{- \frac{l}{c}}\quad \sin \quad \psi \quad {\psi}}} & (27)\end{matrix}$

Rearranging to find the error in angle (dψ) with error in delta:$\begin{matrix}{\frac{\psi}{{\Delta}\quad T} = {- \frac{c}{l\quad \sin \quad \psi}}} & (28)\end{matrix}$

This error function is a minimum when ψ is at 90 degrees, when thesource is directly broadside to the baseline. The function has asingularity at ψ equal to 0 or 180 degrees. A weighing function isselected that gives more weight to baseline measurements taken whendelta T is very small and giving little weight to measurements wheredelta T is near its maximum value (l/c). This relationship is expressedin terms of variances by squaring: $\begin{matrix}{\sigma_{\psi}^{2} = {\frac{c^{2}}{l^{2}\quad \sin^{2}\quad \psi}\sigma_{\Delta \quad T}^{2}}} & (29)\end{matrix}$

By using a trigonometric identity, the angle ψ is eliminated and delta Tis used instead: $\begin{matrix}{\sigma_{\psi}^{2} = \frac{\sigma_{\Delta \quad T}^{2}}{\frac{l^{2}}{c^{2}} - {\Delta \quad T^{2}}}} & (30)\end{matrix}$

When combining measurements from various baselines, it is desired toweight the results to favor those baselines with lesser errors. To dothis, a weight matrix W is introduced. This matrix has the propertiesthat it is diagonal (which assumes measurements are uncorrelated) andthat its trace is equal to one (does not change the value of the finalresult). Assuming that the basic errors in timing determination are thesame for each baseline, a fixed value to the timing variance isassigned. The measurements can be weighed as follows: $\begin{matrix}{W = {\frac{1}{\sum\limits_{i = 1}^{3}\quad \frac{1}{\sigma_{\psi i}^{2}}}\begin{pmatrix}\frac{1}{\sigma_{\psi 1}^{2}} & 0 & 0 \\0 & \frac{1}{\sigma_{\psi 2}^{2}} & 0 \\0 & 0 & \frac{1}{\sigma_{\psi 3}^{2}}\end{pmatrix}}} & (31)\end{matrix}$

This weight matrix is employed to obtain a weighted least squareestimate by multiplying both sides of the model equation thenrearranging into the generalized inverse. This proceeds as follows:

WδΔT=WHΔΘ  (32)

Solving for the angle adjustments:

ΔΘ=(H ^(T) WH)⁻¹ H ^(T) WδΔT  (33)

Using this equation, the measurement variances can be propagated intothe covariance matrix of the AZ, EL angles solution. The covariancematrix of measurements is composed of the variances of the measurementson the diagonal and the covariances or correlations off diagonal asfollows: $\begin{matrix}{{{COV}\left( {\Delta \quad T_{i}} \right)} = {{E\left\{ {ɛ_{\Delta \quad {Ti}}ɛ_{\Delta \quad {Ti}}^{T}} \right\}} = {\Sigma_{\Delta \quad T} = \begin{pmatrix}\sigma_{1}^{2} & \sigma_{12} & \sigma_{13} \\\sigma_{21} & \sigma_{2}^{2} & \sigma_{23} \\\sigma_{31} & \sigma_{32}^{2} & \sigma_{3}^{2}\end{pmatrix}}}} & (34)\end{matrix}$

Since it is assumed that the errors are the same for each baseline andthey are not correlated, this reduces to the identity matrix times asingle measurement covariance:

Σ_(ΔT)=σ_(ΔT) ² I  (35)

The covariance matrix of the angle solution is found by using theweighted generalized inverse equation above:

 Σ_(Θ)=(H ^(T) WH)⁻¹ H ^(T) WΣ _(ΔT)  (36)

The covariance matrix for the angles is given by the following:$\begin{matrix}{\Sigma_{\Theta} = \begin{pmatrix}\sigma_{AZ}^{2} & \sigma_{ELAZ} \\\sigma_{AZEL} & \sigma_{EL}^{2}\end{pmatrix}} & (38)\end{matrix}$

This matrix is positive symmetric and therefore defines an ellipse onthe sky centered around the estimated source point. The ellipse hassemimajor and semiminor axes and inclination to the azimuth axis aredefined by: $\begin{matrix}{a^{2} = {{\frac{1}{2}\quad \left( {\sigma_{AZ}^{2} + \sigma_{EL}^{2}} \right)} + \sqrt{{\frac{1}{4}\quad \left( {\sigma_{AZ}^{2} - \sigma_{EL}^{2}} \right)^{2}} + \sigma_{AZEL}^{2}}}} & (39) \\{b^{2} = {{\frac{1}{2}\quad \left( {\sigma_{AZ}^{2} + \sigma_{EL}^{2}} \right)} - \sqrt{{\frac{1}{4}\quad \left( {\sigma_{AZ}^{2} - \sigma_{EL}^{2}} \right)^{2}} + \sigma_{AZEL}^{2}}}} & (40) \\{{\tan \quad 2\quad \gamma} = \frac{2\quad \sigma_{AZEL}}{\sigma_{AZ}^{2} - \sigma_{EL}^{2}}} & (41)\end{matrix}$

Note that the angular portion of the sky is being treated as beinglocally Cartesian which is valid since this is a small patch of sky. Ifthe input values for timing error represent one standard deviation of arandom sampling of such timing errors then the sigma's above alsorepresent a similar confidence region (i.e. 68.3% of expected eventsoccur with ±σ of the estimated point. However, the ellipse, because itrepresents cross correlations will enclose only 39.4% of events for thesame 1 sigma input. This low confidence level may cause some users toutilize an ellipse enclosing a 95% confidence limit, which requires useof 2.447 times one sigma for the measurement input.

The tools presented herein allow one to compute the error ellipseproperties for all portions of the sky given the configuration of thebaselines (lengths, angles, timing errors) and thereby compare differentconfigurations of detection systems and perform studies regardingaccuracies required and so on. This also allows the computation of theexpected accuracy for specific detected events and to describe the errorellipse onto an image of the scene to assist in identifying the locationof the lightning attachment.

A question arises regarding the change in apparent direction to thesource caused by the wind. This effect could be serious in the case ofthe present single station system. An embodiment of the sensor array ismade of individual detectors spaced a distance D apart, where D is acharacteristic dimension rather than an exact dimension (for estimationpurposes). At a range R, a distance D subtends an angle dθ. Since dθ isvery small:

D≅R·dθ  (42)

The following analysis shows that the wind does not significantly affectthe measured angle of arrival. If the velocity as a function of θ issufficient to cause an additional difference in time of arrival betweenthe two sensors then the derived angle to the source will be incorrect.By taking the derivative of the velocity expression as a function ofangle θ, the time difference measured can be derived. The derivative ofV with angle is: $\begin{matrix}{\frac{V}{\theta} = {V_{W}\quad \sin \quad \left( {\theta - \vartheta} \right)}} & (43)\end{matrix}$

With an expression for dθ in terms of R and D: $\begin{matrix}{{\Delta \quad V} = {\frac{D \cdot V_{W}}{R}\quad \sin \quad \left( {\theta - \vartheta} \right)}} & (44)\end{matrix}$

Applying this expression to the expression involving ΔT and velocity,the additional ΔT or the δΔT as a function of array dimension, range andwind velocity can be estimated as: $\begin{matrix}{{\delta \quad \Delta \quad T} = {{{- \frac{R}{V^{2}}}\quad \Delta \quad V} \cong \frac{D \cdot V_{W}}{c^{2}\quad \sin \quad \left( {\theta - \vartheta} \right)}}} & (45)\end{matrix}$

With an array dimension of one meter, a wind speed of 50 mph (22.6 m/s),a range of 100 meters, and the sine function equal to one (wind alongdirection to source); a timing skew of only 0.00025 seconds is found.This can be translated into angular error by remembering that for abaseline D between two receivers that measure a time of arrivaldifference of ΔT, the angular error caused by an error in ΔT is:$\begin{matrix}{{\psi} = {\left( \frac{c}{D\quad} \right)\quad \frac{- 1}{\sin \quad \psi}\quad \delta \quad \Delta \quad T}} & (46)\end{matrix}$

The angle ψ is used to differentiate this angle from the azimuth to thesource (which is derived from a number of baseline angles). Inserting inthe value for ψ as 45 degrees (sine equals 0.7071) and using theprevious numbers for timing error and so on, an angle error of 0.106radians or 0.0019 degrees or 6.7 seconds of arc is determined.

In the above example, the range error was roughly equal to the ratio ofthe wind speed to the sound speed which is significant. For the case ofangular determination, the error was so small that it can be neglected.This means that the single station system may require and externalsource of wind speed and direction to correct the measured range values(or be accepted as a known systematic error).

In an alternate method, using an array of three baselines of equallength and mutually orthogonal (such that we have a baseline along ourx, y, and z axes), an additional equation of constraint can be provided.The following sections indicate potential uses for that equation inassisting in the solution of direction finding problems. Equation 2translates the ΔT measurements into the direction cosines between eachaxis and the vector pointing to the source. Remembering that thedirection cosines for an orthogonal array follow the relationship:

1=cos² α+cos² β+cos² γ  (47)

where α, β, and γ are the angles between the vector pointing to thesource s and the x, y, and z baselines respectively. Equation 47 is usedto perform a quick validation of the raw ΔT measurements, which is a keyadvantage of an orthogonal array. For orthogonal baselines:

Baseline Cos EL₀₁ Sin EL₀₁ Cos AZ₀₁ Sin AZ₀₁ 01 (x) 1 0 0 1 02 (y) 1 0 10 03 (z) 0 1 0 0

Using these values in equation 17 results in three measurement equationsrelating the ΔT's to AZ and EL:

(T ₀ −T ₁)=cos EL sin AZ

(T ₀ −T ₂)=cos EL cos AZ

(T ₀ −T ₃)=sin EL  (48)

Note that it is assumed that all of the l's are equal and l/c=1. Thesubscript “s” on the angles is also dropped. At the end of thederivations, the l/c coefficient is reinserted. To further simplify,“cos EL” is written as “CEL”, etc. To solve (48), a minimumleast-squared-error approach is used to find AZ and EL so that:

Σ(ΔT _(MEAS) −ΔT _(THEORY))²=Minimum  (49)

Each of the terms in parenthesis are call “residuals.” To find thesolution for AZ, a partial derivative of (49) is taken with respect toAZ and set to zero: $\begin{matrix}{{\frac{\partial}{\partial{AZ}}\left\{ {\left( {{\Delta \quad T_{01}} - {CELSAZ}} \right)^{2} + \left( {{\Delta \quad T_{02}} - {CELCAZ}} \right)^{2} + \left( {{\Delta \quad T_{03}} - {SEL}} \right)^{2}} \right\}} = 0} & (50)\end{matrix}$

Evaluating this expression: $\begin{matrix}{{AZ} = {\tan^{- 1}\left( \frac{\Delta \quad T_{01}}{\Delta \quad T_{02}} \right)}} & (51)\end{matrix}$

Note that the arc tangent has ambiguities due to multiple values of AZwith the same arc tangent. Solving for EL: $\begin{matrix}{{\frac{\partial}{\partial{EL}}\left\{ {\left( {{\Delta \quad T_{01}} - {CELSAZ}} \right)^{2} + \left( {{\Delta \quad T_{02}} - {CELCAZ}} \right)^{2} + \left( {{\Delta \quad T_{03}} - {SEL}} \right)^{2}} \right\}} = 0} & (52)\end{matrix}$

and evaluating as above: $\begin{matrix}{{EL} = {\tan^{- 1}\left\{ \frac{\Delta \quad T_{03}}{{\Delta \quad T_{01}{SAZ}} + {\Delta \quad T_{02}{CAZ}}} \right\}}} & (53)\end{matrix}$

The equations contain a couple of interesting results. First, equation(51) for AZ contains no dependence on ΔT₀₃. This is because ΔT₀₃ gives aline of position across the sky that is at a fixed EL and has nodependence on AZ.

Each measurement provides an arc that makes a large circle on the sky.If the measurements contain some errors, the three circles will notintersect at a single point but will form a triangle as shown in FIG. 7.Each corner of the triangle represents the AZ and EL solution foundusing only two of the three baselines. The question is where in AZ andEL is the “best” or most probable solution given the measured ΔT's?

Equations (41) and (42) provide the least squared error (i.e., the sumof the distances between the chosen AZ, EL point and the lines ofposition is a minimum). Unfortunately, this procedure is influencedstrongly by a measurement that is in error so is not optimum.

Measurement errors consist of random and bias errors. Random errors aredue to noise and other random processes in the measurement chain. Randomerrors are typically minimized by averaging a number of measurements.This is not possible in a lightning measurement system since the eventonly occurs once. Bias errors can, at least in theory, be eliminatedthrough a process of calibration and affect, in some consistent way, allmeasurements. Bias errors that are unknown are generally treated asrandom since they are unknown.

To find how each measurement may be affected by random errors, equation(6) is examined and the derivatives of both sides are taken:

 cdΔT=−l sin ψdψ  (54)

Rearranging: $\begin{matrix}{\frac{\psi}{{\Delta}\quad T} = {\left( \frac{c}{l} \right)\quad \frac{- 1}{\sin \quad \psi}}} & (55)\end{matrix}$

This expression can be interpreted as the ratio of standard deviations.Taking the absolute value (since the timing error can be positive ornegative): $\begin{matrix}{\sigma_{\psi} = {\left( \frac{c}{l} \right)\quad \frac{1}{\sin \quad \psi}\quad \sigma_{\Delta \quad T}}} & (56)\end{matrix}$

If the standard deviation of the timing difference error is 1 μs, thebaseline length is 2 meters, and using a value of c equal to 330 m/s, wesee that: $\begin{matrix}{\sigma_{\psi} = {\frac{(0.000165)}{\sin \quad \psi} \times \left( \frac{180}{\pi} \right)}} & (57)\end{matrix}$

Where the last term converts from radians to degrees. The best accuracyis achieved when the source is at right angles to the baseline and isequal to 0.009 degrees (in the absence of other errors). Note that thisis roughly ½ an arc minute.

FIG. 8 shows the angular accuracy as a function of ψ (“delta psi denotesσψ the standard deviation of ψ). This indicates that, in areas where agiven baseline is parallel to the incoming rays, it should probably beexcluded from the solution.

One embodiment of a data processing algorithm is provided below, withreference to FIG. 9, which results in an estimate of the range, azimuth,and elevation of the source. The system is placed in a ready state 200and waits for an electric field trigger event 202. A timer is started204 upon trigger and the system waits for a sonic wave to be detected206. If sonic data is not received by a predetermined threshold time,such as 150 μsec, the timer is stopped 212 and the system reset 230. Ifsonic data is received 206, the data is filtered for distant strikes 208and the timer is stopped 210. A range for the strike is calculated 214and the sonic data is cross-correlated 216. The range and soniccalculations can be adjusted for environmental conditions 218, such aswind, temperature and humidity. The data is validated 220 by taking thesum of the squares equal to “1” (equation 17). If the data is not valid,the system is reset. All ΔT's are validated 221 that they fall below“0.9.” If valid, AZ and EL are calculated 222 using equations 21 and 22.If not valid, high ΔT measurements are excluded and the AZ and EL arecalculated 224 using remaining equations in (18). AZ and EL errors arecalculated 226 by setting values of ΔT higher and lower by 1 sigma, andAZ and EL are recomputed. AZ±ΔAZ, EL±ΔEL, R±ΔR are output 228 and thesystem is reset 230 for a next event.

The above algorithm is one embodiment of a methodology to approximatethe location of a lightning strike, other methods can be implemented andthe invention is not limited to the above methodology. Although specificembodiments have been illustrated and described herein, it will beappreciated by those of ordinary skill in the art that any arrangement,which is calculated to achieve the same purpose, may be substituted forthe specific embodiment shown. This application is intended to cover anyadaptations or variations of the present invention. Therefore, it ismanifestly intended that this invention be limited only by the claimsand the equivalents thereof.

What is claimed is:
 1. A system to determine an approximate location oflightning strikes comprising: a single electric field sensor; aplurality of acoustic sensors approximately collocated with the electricfield sensor; and a processor to determine a distance to lightningstrikes based on a time differential between an electric field pulse andan acoustic wave at one of the plurality of acoustic sensors, and anangle of arrival based on a difference of time of arrival of the sonicwave at the plurality of acoustic sensors, wherein the processor outputsa range, azimuth and elevation to approximate the location of thelightning strike relative to the location of the system.
 2. The systemof claim 1 wherein the processor compensates for a speed of sound basedon a measured temperature and/or humidity.
 3. The system of claim 1wherein the processor compensates for the time of arrival of the sonicwave using a measured wind speed.
 4. The system of claim 1 wherein theprocessor provides an estimate of the accuracy of the approximatelocation.
 5. The system of claim 1 wherein the plurality of acousticsensors comprise five acoustic sensors.
 6. The system of claim 5 whereinfour of the five acoustic sensors are located on a perimeter of a circleon a horizontal plane, the electric field sensor is located inside thecircle, and a remaining one of the five acoustic sensors is locatedvertically above the circle.
 7. The system of claim 5 wherein four ofthe five acoustic sensors are located on a common horizontal plane. 8.The system of claim 7 wherein the electric field sensor is located onthe common plane and between the four acoustic sensors.
 9. The system ofclaim 8 wherein a remaining one of the five acoustic sensors is locatedvertically above the common plane.
 10. The system of claim 6 wherein thecircle has a radius of about 5 meters and the one acoustic sound sensoris located about two meters above the electric field sensor.
 11. Asystem to determine an approximate location of lighting strikescomprising: an electric field sensor; a plurality of acoustic soundsensors collocated with the electric field sensor; a temperature sensor;and a processor to determine a distance to lighting strikes based on atime differential between an electric field pulse and an acoustic waveat one of the plurality of acoustic sensors, and an angle of arrivalbased on a difference of a time of arrival of the sonic wave at theplurality of acoustic sensors, the processor uses a look-up table tocompensate for a speed of sound based on ambient temperature measured bythe temperature sensor.
 12. The system of claim 11 wherein the pluralityof acoustic sound sensors comprise five acoustic sensors, wherein fourof the five acoustic sound sensors are located on a perimeter of acircle on a horizontal plane, the electric field sensor is located abovethe circle, and a remaining one of the five acoustic sound sensors islocated vertically above the circle.
 13. The system of claim 11 whereinthe processor further compensates for the speed of sound based on ameasured humidity.
 14. The system of claim 11 wherein the processorcompensates for the time of arrival of the sonic wave using a measuredwind speed.
 15. The system of claim 11 wherein the processor outputs arange, azimuth, and elevation relative to the system.
 16. The system ofclaim 15 wherein the processor further outputs an estimate of theaccuracy of the range, azimuth, and elevation.
 17. A method ofdetermining an approximate location of a lightning strike comprising thefollowing steps: measuring the ambient temperature; collecting lightningstrike information, including a time of arrival of an electric fieldpulse using an electric field sensor and a time of arrival data of anassociated sonic wave from a lightning strike, wherein the sonic wavedata is detected using a plurality of acoustic sensors collocated withthe electric field sensor; and processing the lightning strikeinformation including compensating for a speed of sound based on theambient temperature approximate the location of the lightning strike.18. The method of claim 17 wherein compensating comprises accessing alook-up table.
 19. The method of claim 17 wherein processing thelightning strike information comprises calculating an estimate of theaccuracy of the approximate location of the lightning strike.
 20. Themethod of claim 17 further comprises: measuring a wind speed;compensating for the time of arrival of the acoustic wave based onmeasured wind speed while processing the lightning strike information.21. The method of claim 17 wherein a processor outputs a range, azimuth,and elevation relative to the sensors.
 22. The method of claim 21wherein the processor further outputs an estimate of the accuracy of therange, azimuth, and elevation.
 23. The method of claim 22 furthercomprises indicating an estimate of the accuracy of the range, azimuth,and elevation.
 24. A method for determining an approximate location of alightning strike comprising the following steps: detecting an electricfield signal created by the lightning strike with a single electricfield sensor; detecting a sonic pulse created by the lightning strikewith a plurality of approximately collocated acoustic sensors; measuringa time of arrival difference between the electric field signal and thesonic pulse at one of the plurality of approximately collocated sensorsto determine an approximate distance from the lightning strike to theelectric field sensor; measuring a time of arrival difference betweenthe sonic pulse at each of the plurality of approximately collocatedacoustic sensors to determine an angle of arrival of the sonic pulse;and providing a range, azimuth and elevation of the location of thelightning strike relative to the electric field sensor.
 25. The methodof claim 24 further comprises compensating for a speed of sound based ona measured temperature and/or humidity.
 26. The method of claim 24further comprises compensating for the time of arrival of the sonicpulse based on a measured temperature and/or humidity.
 27. A method ofdesigning a single station lightning location system comprising;positioning a single electric field sensor; defining at least a firstgeometry for locating a plurality of acoustic sensors about the singleelectric field sensor; calculating an accuracy of direction of arrivalestimates based on the at least first selected geometry; defining asecond geometry for locating the plurality of acoustic sensors about thesingle electric field sensor; calculating an accuracy of direction ofarrival estimates based on the second selected geometry; and comparingthe accuracy of direction and arrival estimates based on the first andsecond geometries with the expected location of the lightning strike todetermine the geometry providing the greatest accuracy in determiningthe range, azimuth and elevation of the lightning strike as compared tothe location of the electric field sensor.